# Enzyme inhibition

Binding sites of an inhibitor shown in simplified form in the case of a competitive (left) or non-competitive (right) enzyme inhibition ( E enzyme, I inhibitor, S natural substrate)

Enzyme inhibition (also enzyme inhibition ) is the inhibition of an enzymatic reaction by an inhibitor, which inhibitor is mentioned. The rate of the catalyzed reaction is thereby reduced. Inhibitors can bind to different reactants , such as the enzyme or the substrate . The binding site on the enzyme can also vary from the active site to which the substrate binds to other sites that are important for the activity of the enzyme . Enzyme inhibition plays an important role in regulating metabolism in all living things.

## Basics

Enzymes are essential for every organism . They are involved in every metabolic process and act as catalysts for most reactions. In order to be able to regulate these processes, the cells need certain mechanisms that influence the activity of the enzymes. Some enzymes can be switched on by modifications , i.e. activated. For example, the pyruvate kinase required in the utilization of glucose is regulated by phosphorylation ; That is, a phosphoryl group can be attached to the enzyme. This phosphorylated form of pyruvate kinase is not very active. However, if the enzyme has not been modified by a phosphoryl group, it is fully active.

The activity of enzymes can also be influenced by the binding of certain substances. These substances are called effectors . Depending on how effectors act on an enzyme, they are called activators or inhibitors. Activators increase the activity of enzymes, i. that is, they promote the reaction catalyzed by the enzyme. Inhibitors lower the activity and thus inhibit the reaction catalyzed by the enzyme.

There are other ways of reducing enzyme activity, but these do not belong to enzyme inhibition. These include influences from temperature , pH value , ionic strength or solvent effects . These factors have an unspecific effect on a large number of processes.

## Classification of enzyme inhibition

The enzyme inhibition is divided into reversible and irreversible inhibition , depending on the binding of the inhibitor . In the case of reversible enzyme inhibition, the inhibitor can be split off or displaced by the enzyme again. It does not bind tightly to the enzyme. This form of enzyme inhibition is used to regulate various metabolic processes that should not take place temporarily. For example, glycolysis is used to generate energy from glucose. One enzyme involved in this process is the aforementioned pyruvate kinase. Another glycolysis enzyme is phosphofructokinase . If the cell has a lot of energy, it stores it in the form of adenosine triphosphate (ATP). As an inhibitor, this ATP inhibits both phosphofructokinase and pyruvate kinase. This means that no more glucose is converted into energy, i.e. ATP. This special form of enzyme inhibition, in which the end product retroactively inhibits the generating enzyme, is called end product inhibition or feedback inhibition .

With irreversible inhibition, the inhibitor binds so tightly that it can no longer be detached from the enzyme. The activity of the enzyme is lost. Antibiotics , for example, develop irreversible inhibition of some fungi to protect them.

## Reversible enzyme inhibition

Fig. 1: General mechanism of enzyme inhibition (E = enzyme, S = substrate, P = product, I = inhibitor, ES = enzyme-substrate complex, EI = enzyme-inhibitor complex, ESI = enzyme-substrate-inhibitor complex ,  =
Rate constants )${\ displaystyle k_ {x}}$
Fig. 2: Overview of different mechanisms of reversible inhibition

In the case of reversible enzyme inhibition, the inhibitor I binds reversibly to the enzyme E and thus lowers its activity or the rate of reaction of the substrate S to the product P. The inhibitor can, however, for example be displaced by the substrate again. From a mathematical point of view, the rate constant indicates the rate of the uninhibited reaction. If the inhibitor now binds to the enzyme-substrate complex ES, the reaction is defined by the rate constant . is less than . Since the enzyme inhibition is reversible, the reactions to ES, the enzyme-inhibitor complex EI and the enzyme-substrate-inhibitor complex ESI are in equilibrium . All rate constants also for the equilibria are shown in Fig. 1 by the corresponding reaction arrows. ${\ displaystyle k_ {2}}$${\ displaystyle k_ {6}}$${\ displaystyle k_ {6}}$${\ displaystyle k_ {2}}$

In steady state , the reaction rate can even be determined mathematically:

${\ displaystyle {\ frac {d [\ mathrm {P}]} {dt}} = k_ {2} [\ mathrm {ES}] + k_ {6} [\ mathrm {ESI}] = v}$

By reformulating the following equation for the reaction rate, which is generally valid for reversible enzyme inhibition:

${\ displaystyle v = {\ frac {\ left (V_ {1} + {\ frac {V_ {2} [\ mathrm {I}]} {K_ {iu}}} \ right) [\ mathrm {S}] } {K_ {m} \; \ left (1 + {\ frac {[\ mathrm {I}]} {K_ {ic}}} \ right) + \ left (1 + {\ frac {[\ mathrm {I }]} {K_ {iu}}} \ right) \; [\ mathrm {S}]}}}$

The introduced constant corresponds and the constant . The equilibrium constants and can be derived from Fig. 1: and . ${\ displaystyle V_ {1}}$${\ displaystyle k_ {2} [\ mathrm {E}] _ {0}}$${\ displaystyle V_ {2} = k_ {6} [\ mathrm {E}] _ {0}}$${\ displaystyle K_ {ic}}$${\ displaystyle K_ {iu}}$${\ displaystyle K_ {ic} = k _ {- 3} / k_ {3}}$${\ displaystyle K_ {iu} = k _ {- 4} / k_ {4}}$

In Fig. 2 the most important inhibiting mechanisms of the reversible enzyme inhibition are listed. The division was made into full and partial inhibition, which differ in terms of. In the case of complete enzyme inhibition , this value is 0; in the case of partial inhibition , this value is not equal to 0. This means that the enzyme retains its catalytic activity in the case of partial inhibition , although this is influenced by the inhibitor. When the enzyme is completely inhibited, however, the ESI complex can no longer participate in the reaction and is therefore inactive. ${\ displaystyle V_ {2}}$${\ displaystyle V_ {2}}$

Product inhibition is a special case of competitive inhibition because the inhibitor is the product of the reaction. The substrate excess inhibition is a special case of the uncompetitive inhibition. In this case, the substrate is the inhibitor if it is present in high concentration.

### Competitive inhibition

Fig. 3: Mechanism of competitive inhibition (E = enzyme, S = substrate, P = product, I = inhibitor, ES = enzyme-substrate complex, EI = enzyme-inhibitor complex)
Fig. 4: Dependence of the initial speed ( ) on the starting substrate concentration ( ) of a competitive inhibition with (2) and without (1) addition of inhibitor${\ displaystyle v}$${\ displaystyle [\ mathrm {S}] _ {0}}$
Fig. 5: Double reciprocal plots of competitive inhibition in the presence of different inhibitor concentrations (1: 2:  3:  )${\ displaystyle [\ mathrm {I}] = 0}$${\ displaystyle [\ mathrm {I}]> 0}$${\ displaystyle [\ mathrm {I}]> [\ mathrm {I_ {von2}}]}$

Competitive inhibitors are substances that compete with the substrate for the binding site in the active center of the enzyme. They are not converted and can therefore be displaced by the substrate again. Competitive inhibitors are often very similar to the substrate. The mechanism of competitive inhibition is shown in Figure 3. It can be clearly seen that the enzyme E cannot bind the substrate S and the inhibitor I at the same time. The reversible binding of S or I to E creates an equilibrium between the free enzyme E, the enzyme-substrate complex ES and the enzyme-inhibitor complex EI. Assuming that both the substrate S and the inhibitor I are present in much higher concentrations than the enzyme E, the following reaction rate equation can be formulated for the steady-state state:

${\ displaystyle v = {\ frac {V _ {\ max} \; [\ mathrm {S}] _ {0}} {[\ mathrm {S}] _ {0} + K_ {m} \ cdot i}} }$

The constants contained are defined as follows:

${\ displaystyle V _ {\ max} = k_ {2} \, [\ mathrm {E}] _ {0}, \ qquad K_ {m} = {\ frac {(k _ {- 1} + k_ {2}) } {k_ {1}}}}$

and

${\ displaystyle i = 1 + \ left ({\ frac {k_ {3} [\ mathrm {I}] _ {0}} {k _ {- 3}}} \ right)}$.

The reaction rate is shown in Fig. 4 as a function of the starting substrate concentration in the absence and presence of the competitive inhibitor. The Michaelis-Menten constant is increased by the factor in the presence of the inhibitor . However, the maximum speed remains unchanged. ${\ displaystyle [\ mathrm {S}] _ {0}}$ ${\ displaystyle K_ {m}}$${\ displaystyle i}$${\ displaystyle V _ {\ max}}$

The linearization of the Michaelis-Menten plot is achieved by the reciprocal of the equation.

${\ displaystyle {\ frac {1} {v}} = {\ frac {1} {V _ {\ max}}} + {\ frac {K_ {m} \ cdot i} {V _ {\ max}}} \ ; {\ frac {1} {[\ mathrm {S}] _ {0}}}}$

The double reciprocal plot, i.e. the plot of versus in the presence of different concentrations of the inhibitor, is shown in FIG. 5 and is called the Lineweaver-Burk plot . The slope of the graph increases in the presence of the inhibitor . Since remains unchanged, all straight lines on the ordinate intersect at the point . The points of intersection with the abscissa reflect the value . ${\ displaystyle 1 / v}$${\ displaystyle 1 / [\ mathrm {S}] _ {0}}$${\ displaystyle i}$${\ displaystyle V _ {\ max}}$${\ displaystyle 1 / V _ {\ max}}$${\ displaystyle -1 / K_ {m} i}$

The equations described above were derived for the competitive inhibition in which the simultaneous binding of substrate and inhibitor is excluded. However, the conditions for competitive inhibition can also be met if the inhibitor does not occupy the same binding site on the enzyme as the substrate. Binding in the active center, which sterically restricts substrate binding, also leads to the competitive inhibitor type.

#### Inhibition by a competing substrate

In this special case of reversible enzyme inhibition, the enzyme is able to catalyze two reactions, i.e. to bind two different substrates. The enzyme E binds the substrate A and converts it to the product P. The binding of the substrate B to the enzyme E leads to the formation of product Q. Thus, the reaction rate is reduced by adding the substrate B, since the second substrate B can also be bound and converted. In this competitive reaction, one substrate acts on the reaction rate of the reaction of the other substrate as a competitive inhibitor. This results in the following speed equations: ${\ displaystyle v_ {A}}$

${\ displaystyle v_ {A} = {\ frac {V _ {\ max A} \; [A] _ {0}} {[A] _ {0} + K_ {mA} \ cdot i_ {B}}} \ qquad v_ {B} = {\ frac {V _ {\ max B} [B] _ {0}} {[B] _ {0} + K_ {mB} \ cdot i_ {A}}}}$
${\ displaystyle i_ {B} = \ left (1 + {\ frac {[B] _ {0}} {K_ {mB}}} \ right) \ qquad \ qquad i_ {A} = \ left (1+ { \ frac {[A] _ {0}} {K_ {mA}}} \ right)}$

### Non-competitive and allosteric inhibition

With allosteric inhibition (Greek: allos: different; steros: place), the inhibitors, also called allosteric effectors , are not attached to the active center (as is the case with competitive inhibition), but to another point on the enzyme, the allosteric one Center . The conformation of the enzyme is changed in such a way that the binding of the substrate to the active center is made difficult or even impossible.

The allosteric inhibition can only be reversed by removing the effector. An enzyme that catalyzes the first reaction in a chain of reactions is often inhibited by the substance formed at the end (this case is called end product inhibition / feedback inhibition / negative feedback).

Fig. 6: Mechanism of non-competitive inhibition (E = enzyme, S = substrate, P = product, I = inhibitor, ES = enzyme-substrate complex, EI = enzyme-inhibitor complex, ESI = enzyme-substrate-inhibitor complex ,  =
Rate constants )${\ displaystyle k_ {x}}$
Fig. 7: Dependence of the initial speed ( ) on the starting substrate concentration ( ) of a non-competitive inhibition with (2) and without (1) the addition of inhibitor${\ displaystyle v}$${\ displaystyle [\ mathrm {S}] _ {0}}$
Fig. 8: Double reciprocal plot of a non-competitive inhibition in the presence of different inhibitor concentrations (1: 2:  3:  )${\ displaystyle [\ mathrm {I}] = 0}$${\ displaystyle [\ mathrm {I}]> 0}$${\ displaystyle [\ mathrm {I}]> [\ mathrm {I_ {von2}}]}$

In this non-competitive inhibition, the binding of the inhibitor I to the enzyme E does not affect the substrate binding. The inhibitor I is thus able to bind both to the free enzyme E and to the enzyme-substrate complex ES, i. i.e., the inhibitor does not bind in the substrate-binding part of the enzyme, the active site. The substrate can also react with the enzyme-inhibitor complex EI, but the enzyme-inhibitor-substrate complex EIS that is formed is not able to split off the product P. The reaction mechanism is shown in more detail in Fig. 6.

A simple rate equation can be derived under steady-state conditions:

${\ displaystyle v = {\ frac {V _ {\ max} \; [\ mathrm {S}] _ {0}} {i \ cdot ([\ mathrm {S}] _ {0} + K_ {m}) }}}$.

The constants and are defined as follows: and . From the speed equation one can deduce that the non-competitive inhibitor reduces the maximum speed by the factor in the presence . The value for the substrate remains unchanged. ${\ displaystyle K_ {m}}$${\ displaystyle i}$${\ displaystyle K_ {m} = (k _ {- 1} + k_ {2}) / k_ {1}}$${\ displaystyle i = 1 + (k_ {3} [\ mathrm {I}] _ {0} / k _ {- 3})}$${\ displaystyle V _ {\ max}}$${\ displaystyle 1 / i}$${\ displaystyle K_ {m}}$

The application according to Lineweaver-Burk takes place with the formation of the reciprocal reaction rate according to this formula:

${\ displaystyle {\ frac {1} {v}} = {\ frac {i} {V _ {\ max}}} + {\ frac {K_ {m} \ cdot i} {V _ {\ max}}} \ ; {\ frac {1} {[\ mathrm {S}] _ {0}}}}$.

The Lineweaver-Burk plot is shown in Fig. 8. The increase in the reciprocal reaction rate after the addition of the non-competitive inhibitor is i times higher than without inhibition. The ordinate intersection of each straight line is included . The straight lines for reactions of different inhibitor concentrations intersect at exactly one point on the abscissa, in the value . ${\ displaystyle i / V _ {\ max}}$${\ displaystyle -1 / K_ {m}}$

In some cases of non-competitive inhibition, the behavior of the inhibitor deviates somewhat from the "normal case". The reaction of the inhibitor with the enzyme then takes place much faster than that of the substrate. At low substrate concentrations, the reduction in maximum speed is not so great. Therefore, the following speed equation results for the conditions and : ${\ displaystyle k _ {- 1} \ ll k_ {2}}$${\ displaystyle K_ {m} = k_ {2} / k_ {1}}$

${\ displaystyle v = {\ frac {V _ {\ max} \; [\ mathrm {S}] _ {0}} {[\ mathrm {S}] _ {0} \ cdot i + K_ {m}}} }$.

Such behavior should be expected when determining the inhibitor type of an inhibitor.

### Uncompetitive inhibition

Fig. 9: General mechanism of the incompetitive enzyme inhibition (E = enzyme, S = substrate, P = product, I = inhibitor, ES = enzyme-substrate complex, ESI = enzyme-substrate-inhibitor complex,  =
rate constant )${\ displaystyle k_ {x}}$
Fig. 10: Dependence of the initial speed ( ) on the starting substrate concentration ( ) of an uncompetitive inhibition with (2) and without (1) the addition of inhibitor${\ displaystyle v}$${\ displaystyle [\ mathrm {S}] _ {0}}$

Occasionally, in addition to the competitive and the non-competitive inhibition, the uncompetitive type of inhibition also occurs. The inhibitor only reacts with the enzyme-substrate complex ES, as can be seen in Fig. 9. The equation for the reaction rate derived for this mechanism under steady-state conditions is as follows:

${\ displaystyle v = {\ frac {V _ {\ max} \; [\ mathrm {S}] _ {0}} {i \ cdot \ left ({\ frac {K_ {m}} {i}} + [ \ mathrm {S}] _ {0} \ right)}}}$

The constants are defined as follows: and . From the Michaelis-Menten plot of an uncompetitive inhibition shown in Fig. 10, one can see that the inhibitor changes both the maximum reaction rate and the value. ${\ displaystyle i = 1 + k_ {3} [\ mathrm {I}] _ {0} / k _ {- 3}}$${\ displaystyle K_ {m} = (k _ {- 1} + k_ {2}) / k_ {1}}$${\ displaystyle K_ {m}}$

By transforming the rate equation into the reciprocal form, the dependence of the reaction rate on the initial concentration of the substrate can be represented in a linearized manner:

${\ displaystyle {\ frac {1} {v}} = {\ frac {i} {V _ {\ max}}} + {\ frac {K_ {m}} {V _ {\ max}}} \; {\ frac {1} {[\ mathrm {S}] _ {0}}}}$

From this formula it can be seen that the increase is independent of the uncompetitive inhibitor, that is, the graphs are parallel to one another for different inhibitor concentrations. The zero represents the value . The straight lines intersect the ordinate at the point . ${\ displaystyle i / K_ {m}}$${\ displaystyle i / V _ {\ max}}$

The uncompetitive inhibition occurs, for example, with oxidases when the inhibitor can only react with a certain oxidation level of the enzyme. Another possibility for an uncompetitive inhibitor is offered by an ordered mechanism , a two-substrate reaction in which the inhibitor occurs in competition with one of the substrates.

### Partly competitive inhibition

Fig. 11: General mechanism of partial enzyme inhibition (E = enzyme, S = substrate, P = product, I = inhibitor, ES = enzyme-substrate complex, EI = enzyme-inhibitor complex, ESI = enzyme-substrate-inhibitor Complex,  =
equilibrium constants )${\ displaystyle K_ {x}}$
Fig. 12: Dependence of the value on the initial inhibitor concentration ( ) of a partially competitive inhibition${\ displaystyle K_ {S}}$${\ displaystyle [\ mathrm {I}] _ {0}}$

By definition, there is actually no partially competitive inhibition, since in competitive inhibition substrate and inhibitor cannot bind to the enzyme at the same time, so there is no EIS complex that could convert the substrate. In the older literature, however, this expression is used for a special case of partially non-competitive inhibition with k cat = k * . In this case the Lineweaver-Burk plot is similar to a competitive escapement, the secondary plot is curved. The term “partially competitive” is therefore mechanistically incorrect and should no longer be used.

### Inhibition of excess substrate

Fig. 13: Michaelis-Menten plot of a substrate excess inhibition (1: K i >> K m , 2: K i = K m )

With some enzymes, very high substrate concentrations can bind a second substrate molecule to the enzyme. The resulting ESS complex is unable to break down into product and enzyme. The rate equation for this reaction mechanism is:

${\ displaystyle v = {\ frac {V _ {\ max}} {1 + {\ frac {K_ {m}} {[\ mathrm {S}] _ {0}}} + {\ frac {[\ mathrm { S}] _ {0}} {K_ {i}}}}}}$

The dissociation constant of the ESS complex was designated as . If the value is much lower than the value, a hyperbolic dependence is obtained in the Michaelis-Menten plot (Fig. 13, curve 1). If these two values ​​are approximately the same or if the value is higher, an optimum curve is created, so to speak (Fig. 13, curve 2). ${\ displaystyle K_ {i}}$${\ displaystyle K_ {m}}$${\ displaystyle K_ {i}}$${\ displaystyle K_ {m}}$

### Inhibition by reaction of an inhibitor with the substrate

With this type of inhibitor, the inhibitor reacts with the substrate, which is then no longer converted by the enzyme. The binding of the inhibitor is regarded as reversible in this inhibition. This reduces the substrate concentration freely accessible to the enzyme : ${\ displaystyle [\ mathrm {S} _ {\ mathrm {eff}}]}$

${\ displaystyle [\ mathrm {S_ {eff}}] = [\ mathrm {S}] _ {0} - [\ mathrm {SI}]}$

In the presence of the inhibitor, the maximum speed is reached at high substrate concentrations. In the Lineweaver-Burk diagram, this mechanism can be differentiated from a competitive inhibitor, since a deviation from linearity occurs.

## Inactivation

Fig. 14: Dependence of the activity on the inactivator concentration of an inactivation (explained in the text)

The irreversible binding of the inactivator to the enzyme reduces the catalytic activity (also incorrectly referred to as “irreversible enzyme inhibition”, but inhibition is always reversible by definition ). A dissociation of the enzyme-inactivator complex into free enzyme and inactivator is not possible; that is, the enzyme remains inactive forever. The activity depends linearly on the inactivator concentration. This dependency can be seen in Fig. 14 (curve 1). Deviating from this curve, curve 2 arises in the figure. The inactivator reacts irreversibly with several groups of different specificities, which leads to titration of the more specific groups that are not actively involved in the catalytic mechanism. The groups that are important for catalysis then react, reducing the activity. On the other hand, the inactivator can enter into a complex with the enzyme which still has little activity. This is the case with curve 3 of the figure.

If the inactivator concentration is considerably higher than that of the enzyme, the rate constant for the reaction of the enzyme with the inactivator can be formulated as a pseudo-first order reaction:

${\ displaystyle - {\ frac {d [\ mathrm {E}]} {dt}} = - {\ frac {dA} {dt}} = k \; [\ mathrm {I}] _ {0} \; [\ mathrm {E}] = k '\; [\ mathrm {E}]}$

${\ displaystyle A}$corresponds to the activity in this equation. The rate constant can be determined from the increase by plotting the logarithm of the activity against the reaction time. ${\ displaystyle k '}$

The rate constant is influenced by the presence of the substrate. This is because this protects the enzyme from being inactivated by the inactivator, which means that the rate constant is lower than without the addition of substrate. In the case of such behavior, the inactivator may bind to a specific group in the active center, i.e. it may have the same binding site as the substrate.

An example of inactivation are the so-called “suicide substrates”, which form a covalent bond with the functional group of the enzyme and thus block it. Inactivation by such substances not only requires a binding site on the enzyme, but also catalytic conversion of the inactivator in the enzyme. Such substances are very specific and cause few side effects when administered as medication (example: penicillin ).